[Background] Meticulous modeling of neutrino-nucleus interactions is essential to achieve the unprecedented precision goals of present and future accelerator-based neutrino-oscillation experiments. [Purpose] Confront our calculations of charged-current quasielastic cross section with the measurements of MiniBooNE and T2K, and to quantitatively investigate the role of nuclear-structure effects, in particular, low-energy nuclear excitations in forward muon scattering. [Method] The model takes the mean-field (MF) approach as the starting point, and solves Hartree-Fock (HF) equations using a Skyrme (SkE2) nucleon-nucleon interaction. Long-range nuclear correlations are taken into account by means of the continuum random-phase approximation (CRPA) framework. [Results] We present our calculations on flux-folded double differential, and flux-unfolded total cross sections off $^{12}$C and compare them with MiniBooNE and (off-axis) T2K measurements. We discuss the importance of low-energy nuclear excitations for the forward bins. [Conclusions] The CRPA predictions describe the gross features of the measured cross sections. They underpredict the data (more in the neutrino than in the antineutrino case) because of the absence of processes beyond pure quasielastic scattering in our model. At very forward muon scattering, low-energy nuclear excitations ($omega < $ 50 MeV) account for nearly 50% of the flux-folded cross section.
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